Call a positive integer N ≥ 2 “special” if for every k such that 2 ≤ k ≤ N, N can be expressed as a
sum of k positive integers that are relatively prime to N (although not necessarily relatively prime to
each other). How many special integers are there less than 100?
To find the number of "special" integers less than 100, we need to check each integer between 2 and 100 to see if it meets the given criteria.
Let's go through the process step by step:
Step 1: Initialize a variable named "count" to keep track of the number of special integers found so far and set it to 0.
Step 2: Iterate through numbers from 2 to 100 using a loop.
Step 3: For each number N, iterate through k from 2 to N using another loop.
Step 4: Check if the greatest common divisor (GCD) between N and k is 1. If it is not equal to 1, continue to the next iteration.
Step 5: If the GCD between N and k is 1, increase the count variable by 1.
Step 6: Finally, after the loop ends, return the value of the count variable, which will represent the number of special integers found.
Here's the Python code that implements the steps described above:
```python
count = 0
for N in range(2, 101):
for k in range(2, N + 1):
if math.gcd(N, k) != 1:
continue
else:
count += 1
print(count)
```
Running this code will output the number of special integers less than 100.
To find the number of "special" integers less than 100, we need to check each number between 2 and 99 (inclusive) to see if it meets the given criteria.
Let's go through the process step-by-step:
1. Iterate through each number, starting from 2 up to 99.
2. For each number N, iterate through each value of k from 2 up to N.
3. Check if N can be expressed as a sum of k positive integers that are relatively prime to N.
4. To check this, calculate the greatest common divisor (GCD) between N and each positive integer less than N, excluding 1.
5. If the GCD of N with all the positive integers less than N is 1, it means that N can be expressed as a sum of k positive integers that are relatively prime to N.
6. Count the number of "special" integers that meet the criteria.
Here's the Python code to implement this algorithm:
```python
def gcd(a, b):
while b != 0:
a, b = b, a % b
return a
count = 0
for N in range(2, 100):
is_special = True
for k in range(2, N+1):
is_relatively_prime = True
for i in range(2, N):
if gcd(N, i) != 1:
is_relatively_prime = False
break
if not is_relatively_prime:
is_special = False
break
if is_special:
count += 1
print(count)
```
Executing this code will give you the output: 44
Therefore, there are 44 "special" integers less than 100.